#10668: Refactor category support for morphisms (Hom is not a functorial
construction!)
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Reporter: nthiery | Owner: nthiery
Type: defect | Status: new
Priority: major | Milestone:
Component: categories | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Hom is currently implemented as a covariant functorial
construction:
{{{
sage: Rings().hom_category()
Category of hom sets in Category of rings
sage: Rings().hom_category().super_categories()
[Category of hom sets in Category of sets]
}}}
The intention was to model the fact that a morphism in a category is
also a morphism in any super category, via the forgetful functor. With
the example above, if A and B are rings, then a ring morphism phi:
A->B is also a set morphism. However, at the level of parents, and as
noted in the documentation of sage.category.HomCategory, this is
mathematically plain wrong: if A and B are rings, then Hom(A,B) in the
category of rings does not coincide with Hom(A,B) in the category of
sets.
I, Nicolas, take full blame for this misfeature; it was just the
shortest route to implement some urgently needed features about
morphisms, and still get that huge chunk of category code done.
Here are some thoughts for a proper implementation:
- Add support for a HomMethods subclass, similar to ElementMethods
and ParentMethods. If Cat is a category, then Cat.ElementMethods
will provide generic methods for morphisms in Cat and any
subcategory. This will require to implement the building of a
hierarchy of abstract classes Cat.method_class. And to either tweak
sage.categories.morphism.Morphism to add inheritance from this
class, as is done in Element and Parent, or to tweak the
implementation of Homset.element_class so that, for H a homset in
Cat, H.element_class would inherit from Cat.method_class besides
the usual stuff (I guess I prefer the later option).
- Reimplement HomCategory. For Cat a category, the purpose of
Cat.hom_category() shall be to provide mathematical information
about its homsets (e.g. that a homset in the category of vector
spaces is also a vector space). It should ignore the super
categories of C in general, except if is a full subcategory (a
homset in the category of finite groups is also a homset in the
category of groups).
- If CatA is a subcategory of catB, add automatic coercion (or just
conversion?) from Hom(A,B, CatA) to Hom(A, B, CatB), modeling the
appropriate forgetful functor. There are too many such coercions
for them to be registered explicitly. So this probably need to be
implemented through a specific CatB._has_coerce_map_from(A) for
homsets.
Some extra work will probably be needed if one wants to handle
mixed morphism arithmetic (e.g. having the sum of an algebra
morphism and a coalgebra morphism return a vector space morphism).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10668>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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